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C₃×#213 #3737

order	144
isoclinism	isoclinic class of #213
automorphism	Aut(G) = C₂²×D₃×D₄ (order 192) · Inn(G) = #67 · Out(G) = C₂³
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂

Aliases

Splittings

Fitting	C₃²×D₄ ⋊ C₂
Sylow	C₃² ⋊ D₈

Subgroups

	subgroup	G/H
center	C₂×C₃	#67
commutator, cosocle	C₃×C₄	C₂²×C₃
Frattini	C₄	C₂×C₃×D₃
Fitting	C₃²×D₄	C₂
socle	C₂×C₃²	D₄
hypercenter	C₃×D₄	D₃
hypocenter	C₃	C₃×D₈
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃ ◁ C₃×C₄ ◁ C₃×D₄

52 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 14
transitive degree	μ_t(G) = 24

A minimal faithful permutation representation — 2 generators on 14 points:

(1,2,3)(4,5,6)(8,11)(9,14)(12,13);(5,6)(7,8)(9,11)(10,13)(12,14)

Conjugacy classes & character table

36 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#213, #63925, #63926, …
maximal normal subgroup of	D₃×#213, #63925, #63926, …
maximal quotient of	C₃×#807, C₃×#809, C₃×#826, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(144, 80);